#7715561

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 44 points:

(2,4)(6,8)(10,11)(13,14,19,24)(15,20,25,35)(16,21,29,27)(17,22,32,26)(18,33,34,23)(28,42,39,36)(30,41,43,37)(31,40,44,38);(1,2,3,4)(15,25)(16,29)(20,35)(21,27)(28,39)(30,43)(36,42)(37,41);(5,6,9,8)(7,11,12,10)(13,15)(14,20)(16,26)(17,21)(18,28)(19,25)(22,29)(23,36)(24,35)(27,32)(30,40)(31,37)(33,42)(34,39)(38,43)(41,44);(5,7,9,12)(6,10,8,11)(13,16)(14,21)(15,26)(17,20)(18,30)(19,29)(22,25)(23,37)(24,27)(28,40)(31,36)(32,35)(33,41)(34,43)(38,39)(42,44);(5,8,9,6)(7,11,12,10)(13,17)(14,22)(15,27)(16,20)(18,31)(19,32)(21,25)(23,38)(24,26)(28,41)(29,35)(30,36)(33,40)(34,44)(37,39)(42,43);(13,18,19,34)(14,23,24,33)(15,28,25,39)(16,30,29,43)(17,31,32,44)(20,36,35,42)(21,37,27,41)(22,38,26,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622757);